Problem with Pelletized Begonia seedlings

My new Tuberous rooted seedlings grew roots from the bottom of the pellet and green from the top of the clay like pellet. When I touch them or bottom watered the trays, they fall over. By this time all of the clayish pellet has disappeared, but the seedling is being held up by the roots...

My questions is: do you think i could push down/replant/rehole the seedling, possibly bury the cotyledons and not rot/kill the seedling?

As long as they have some root extending into the media, they should be fine. Small seedlings like begonia and petunia will do that and lay flat against the soil. I would keep some type of lid on your container to help retain some of the moisture so you don't have to water them as much. Keep the lid vented however, you don't want to get them too wet, just prevent so much evaporation from occurring. I would not try to repot/cover them, it is too easy to damage and kill them when they are so small. Wait until they have a couple of leaves before you repot them.